In this paper, we investigate a cache updating system with a server containing $N$ files, $K$ relays and $M$ users. The server keeps the freshest versions of the files which are updated with fixed rates. Each relay can download the fresh files from the server in a certain period of time. Each user can get the fresh files from any relay as long as the relay has stored the fresh versions of the requested files. Due to the limited storage capacity and updating capacity of each relay, different cache designs will lead to different average freshness of all updating files at users. In order to keep the average freshness as large as possible in the cache updating system, we formulate an average freshness-optimal cache updating problem (AFOCUP) to obtain an optimal cache scheme. However, because of the nonlinearity of the AFOCUP, it is difficult to seek out the optimal cache scheme. As a result, an linear approximate model is suggested by distributing the total update rates completely in accordance with the number of files in the relay in advance. Then we utilize the greedy algorithm to search the optimal cache scheme that is satisfied with the limited storage capacity of each relay. Finally, some numerical examples are provided to illustrate the performance of the approximate solution.